Newspace parameters
| Level: | \( N \) | \(=\) | \( 704 = 2^{6} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 704.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(112.910209148\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 22) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 29.0000 | 0 | 31.0000 | 0 | −230.000 | 0 | 598.000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 704.6.a.j | 1 | |
| 4.b | odd | 2 | 1 | 704.6.a.a | 1 | ||
| 8.b | even | 2 | 1 | 22.6.a.c | ✓ | 1 | |
| 8.d | odd | 2 | 1 | 176.6.a.e | 1 | ||
| 24.h | odd | 2 | 1 | 198.6.a.b | 1 | ||
| 40.f | even | 2 | 1 | 550.6.a.c | 1 | ||
| 40.i | odd | 4 | 2 | 550.6.b.a | 2 | ||
| 56.h | odd | 2 | 1 | 1078.6.a.f | 1 | ||
| 88.b | odd | 2 | 1 | 242.6.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.6.a.c | ✓ | 1 | 8.b | even | 2 | 1 | |
| 176.6.a.e | 1 | 8.d | odd | 2 | 1 | ||
| 198.6.a.b | 1 | 24.h | odd | 2 | 1 | ||
| 242.6.a.a | 1 | 88.b | odd | 2 | 1 | ||
| 550.6.a.c | 1 | 40.f | even | 2 | 1 | ||
| 550.6.b.a | 2 | 40.i | odd | 4 | 2 | ||
| 704.6.a.a | 1 | 4.b | odd | 2 | 1 | ||
| 704.6.a.j | 1 | 1.a | even | 1 | 1 | trivial | |
| 1078.6.a.f | 1 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 29 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(704))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 29 \)
$5$
\( T - 31 \)
$7$
\( T + 230 \)
$11$
\( T + 121 \)
$13$
\( T + 112 \)
$17$
\( T + 1142 \)
$19$
\( T - 612 \)
$23$
\( T + 1941 \)
$29$
\( T + 1192 \)
$31$
\( T + 1037 \)
$37$
\( T + 8083 \)
$41$
\( T + 10444 \)
$43$
\( T + 58 \)
$47$
\( T - 8656 \)
$53$
\( T - 20318 \)
$59$
\( T - 21351 \)
$61$
\( T + 47044 \)
$67$
\( T + 48093 \)
$71$
\( T + 24967 \)
$73$
\( T + 42288 \)
$79$
\( T + 72410 \)
$83$
\( T - 15806 \)
$89$
\( T + 114761 \)
$97$
\( T + 5159 \)
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